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1. A weight which is suspended by means of a helical spring vibrates in the gravitational field of the earth. Find the expression for the period, taking the mass of the spring into account.
Let   m = mass of the suspended body. m' = mass of the spring. p = mass per unit length of the spring. L = length of the spring before the body is suspended. D = increase in the length of the spring due to the weight of the
suspended body.
a = the distance through which the body is pulled down in order to start the vibration.
In Fig. 147 let 0 denote the position of equilibrium, A the lowest position, and B any position of the body. The coordinate in terms of which we want to express the energy of the system must vanish at the position of equilibrium. Therefore we will define the position of the suspended body in terms of its distance from the position of equilibrium. The distance will be considered as positive when measured downwards. Let q denote this distance then the kinetic energy of the suspended body equals | mg2. In order to express the kinetic energy of  the spring in terms of this coordinate let x denote the distance of an element of the spring from the point of suspension. Then the kinetic energy of the entire spring is
o 2
2 dm
= / *-p<fa
2  3
1 m'
Hence the kinetic energy of the entire system is